Conjecture on the Euclidean relative sigma-constant of the solid torus

Let $T$ be a solid torus, i.e., a handlebody with one handle. Define

$$\sigma_{\operatorname{euc}}(M)=\sup_{\Omega}Q(\Omega),$$

where the supremum is taken over all smooth, bounded domains $\Omega\subset\mathbb{R}^3$ whose closure is diffeomorphic to $M$. Let $C$ be the region enclosed by the image of a Clifford torus under stereographic projection. Solid-torus Euclidean sigma-constant conjecture. The invariant $\sigma_{\operatorname{euc}}(T)$ is attained by $C$. This proposes a distinguished extremal domain for the Euclidean relative $\sigma$-constant among domains whose closure is diffeomorphic to a solid torus; whether the conjecture holds is not resolved in the supplied text.